On the isomorphism problem for monoids of product-one sequences

TL;DR

This paper resolves the Isomorphism Problem for monoids of product-one sequences over torsion groups by proving that an isomorphism between $\mathcal{B}(G_1)$ and $\mathcal{B}(G_2)$ forces an isomorphism between the underlying groups $G_1$ and $G_2$. The approach extends divisor-theoretic techniques and leverages the complete integral closures to lift an isomorphism to a free abelian monoid, where the base bijection on the group level is shown to be either a homomorphism or an anti-homomorphism, yielding $G_1\cong G_2$. The paper also derives a corollary: when the commutator subgroups are torsion, isomorphism of the product-one monoids implies isomorphism of the torsion subgroups. Overall, the work generalizes known abelian-case results to the torsion non-abelian setting and strengthens the connections between factorization theory and group structure.

Abstract

Let $G_1$ and $G_2$ be torsion groups. We prove that the monoids of product-one sequences over $G_1$ and over $G_2$ are isomorphic if and only if the groups $G_1$ and $G_2$ are isomorphic. This was known before for abelian groups.

Theorems & Definitions (10)

• proof : Proof of Theorem \ref{['1.1']}
• proof : Proof of A1.
• proof : Proof of A2.
• proof : Proof of A3.
• proof : Proof of A4.
• proof : Proof of A5.
• proof : Proof of A6.
• proof : Proof of A7.
• proof : Proof of A8.
• proof : Proof of Corollary \ref{['1.2']}